The properties of iron under core conditions from ﬁrst principles … · 2012-03-26 · Physics of the Earth and Planetary Interiors 140 (2003) 101–125 The properties of iron

Physics of the Earth and Planetary Interiors 140 (2003) 101–125

The properties of iron under core conditions from firstprinciples calculations

L. Vocadloa, D. Alfè a,b, M.J. Gillanb, G. David Pricea,∗a Research School of Earth Sciences, Birkbeck and University College London, Gower Street, London WC1E 6BT, UK

b Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK

Accepted 11 August 2003

Abstract

The Earth’s core is largely composed of iron (Fe). The phase relations and physical properties of both solid and liquid Feare therefore of great geophysical importance. As a result, over the past 50 years the properties of Fe have been extensivelystudied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (∼6000 K) found in thecore provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the resultsobtained by using different experimental techniques. In the past 15 years quantum mechanical techniques have been applied topredict the properties of Fe. Here we review the progress that has been made in the use of first principles methods in the study ofFe, and focus upon (i) the structure of Fe under core conditions, (ii) the highP melting behaviour of Fe, (iii) the thermodynamicproperties of hexagonal close-packed (hcp) Fe, and (iv) the rheological and thermodynamic properties of highP liquid Fe.© 2003 Elsevier B.V. All rights reserved.

Keywords: Earth’s core; Molecular dynamics; Ab initio methods; Iron

1. Introduction

The fact that the core is largely composed of Fe wasfirmly established as a result ofBirch’s (1952)analysisof mass-density/sound-wave velocity systematics. To-day we believe that the outer core is about 6–10% lessdense than pure liquid Fe, while the solid inner-coreis a few percent less dense than crystalline Fe (e.g.Poirier, 1994a). From cosmochemical and other con-siderations, it has been suggested (e.g.Poirier, 1994b;Allègre et al., 1995; McDonough and Sun, 1995) thatthe alloying elements in the core might include S, O,Si, H and C. It is also probable that the core containsminor amounts of other elements, such as Ni and K.

∗ Corresponding author.E-mail address: [email protected] (G.D. Price).

The exact temperature profile of the core is still con-troversial (e.g.Alfè et al., 2002a), but it is generallyheld that the inner-core is crystallising from the outercore as the Earth slowly cools, and that core tempera-tures are in the range 4000–7000 K, while the pressureat the centre of the Earth is∼360 GPa.

Before a full understanding of the chemically com-plex core is reached, it is necessary to understand theproperties and behaviour at high pressure (P) and tem-peratures (T) of its primary constituent, namely metal-lic Fe. Experimental techniques have evolved rapidlyin the past years, and today using diamond anvil cells(DAC) or shock experiments the study of minerals atpressures up to∼200 GPa and temperatures of a fewthousand Kelvin is possible. These studies, however,are still far from routine, and results from differentgroups are often in conflict (see for example reviews

0031-9201/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.pepi.2003.08.001

102 L. Vocadlo et al. / Physics of the Earth and Planetary Interiors 140 (2003) 101–125

by Poirier, 1994b; Shen and Heinze, 1998; Stixrudeand Brown, 1998; Boehler, 2000). As a result, there-fore, in order to complement these existing experi-mental studies and to extend the range of pressure andtemperature over which we can model the Earth, com-putational mineral physics has, in the past decade, be-come an established and growing discipline.

Within computational mineral physics a variety ofatomistic simulation methods (developed originallyin the fields of solid state physics and theoreticalchemistry) are used. These techniques can be di-vided approximately into those that use some formof interatomic potential model to describe the energyof the interaction of atoms in a mineral as a func-tion of atomic separation and geometry, and thosethat involve the approximate solution of Schrödingerequation to calculate the energy of the mineral speciesby quantum mechanical techniques. For the Earthsciences, the accurate description of the behaviour ofminerals as a function of temperature is particularlyimportant, and computational mineral physics usuallyuses either lattice dynamics or molecular dynamics(MD) methods to achieve this important step. Therelatively recent application of all of these advancedcondensed matter physics methods to geophysics hasonly been made possible by the very rapid advancesin the power and speed of computer processing. Tech-niques, which in the past were limited to the study ofstructurally simple compounds, with small unit cells,can today be applied to describe the behaviour ofcomplex, low symmetry structures (which epitomisemost minerals) and liquids.

In this paper, we will focus on recent studies of Fe,which have been aimed at predicting its geophysicalproperties and behaviour under core conditions. Al-though interatomic potentials have been used to studyFe (e.g.Matsui and Anderson, 1997), many of its prop-erties are very dependent upon a precise descriptionof its metallic nature and can only be modelled accu-rately by quantum mechanical methods. Thus, belowwe outline the essential ab initio techniques used inthe most recent studies of Fe (see alsoStixrude et al.,1998). We then present a discussion of the structureof the stable phase of Fe at core pressures and tem-peratures, its melting behaviour at core pressures, andab initio estimates of the highP/T physical and ther-modynamic properties of both liquid and crystallineiron.

2. Quantum mechanical simulations

Ab initio simulations are based on the descriptionof the electrons within a system in terms of a quantummechanical wave function,ψ, the energy and dynam-ics of which is governed by the general Schrödingerequation for a single electron:

ih∂ψ

∂t= − h2

2m∇2ψ + V(r)ψ (1)

whereV(r) is the potential energy of the system andthe other terms take their usual meaning. In minerals,however, it is necessary to take into account all ofthe electrons within the crystal, so the energy,E, of amany electron wave function,Ψ , is required:

EΨ(r1, r2, . . . , rN)

=(∑

− h2

2m∇2 + Vion + Ve–e

)Ψ(r1, r2, . . . , rN)

(2)

In a confined system, the electrons experience in-teractions between the nuclei and each other. Thisinteraction may be expressed in terms of an ioniccontribution and a Coulombic contribution (the sec-ond and third terms within the brackets). Energyminimisation techniques may be applied in order toobtain the equilibrium structure for the system underconsideration.

Unfortunately, the complexity of the wave function,Ψ , for an N electron system scales asMN , whereMis the number of degrees of freedom for a single elec-tron wave function,Ψ . This type of problem cannotreadily be solved for large systems due to computa-tional limitations, and therefore the exact solution tothe problem for large systems is intractable. However,there are a number of approximations that may bemade to simplify the calculation, whereby good pre-dictions of the structural and electronic properties ofmaterials can be obtained by solving self-consistentlythe one-electron Schrödinger equation for the system,and then summing these individual contributions overall the electrons in the system (for more detailed re-views seeGillan (1997)and Stixrude et al. (1998)).Such approximation techniques include the HartreeFock approximation (HFA) and density functionaltheory (DFT), which differ in their description ofthe electron–electron interactions. In both cases the

L. Vocadlo et al. / Physics of the Earth and Planetary Interiors 140 (2003) 101–125 103

average electrostatic field surrounding each electron istreated similarly, reducing the many body Hamiltonianin the Schrödinger equation for a non-spin-polarisedsystem (i.e. ignoring any spin alignment and resultingmagnetic effects) to that for one electron surroundedby an effective potential associated with the interac-tions of the surrounding crystal. However, the differ-ence between the two methods arises in the treatmentof the contribution to the potential associated withthe fact that the electron is not in an average field,the correlation, and also in the treatment of the elec-tronic spin governed by Pauli’s exclusion principle,the exchange. In the HFA the exchange interactionsare treated exactly, but the correlation is not included;in modern geophysical investigations of silicates andiron phases, DFT is increasingly favoured, where theexchange and correlation are both included but, incurrent approximations, only in an average way (seeStixrude et al. (1998)for more details).

Density functional theory, originally developed byHohenberg and Kohn (1964)and Kohn and Sham(1965), describes the exact ground state properties ofa system in terms of a unique functional of chargedensity alone, i.e.E = E(ρ). The Hohenberg–Kohntheorem says that the ground state density uniquelydetermines the potential (and so all equilibrium phys-ical properties of the system). Using the variationalprinciple it is easy to show that the ground state den-sity minimises the total-energy. So by looking for theminimum of the total-energy, we find the ground statedensity. In DFT, the electronic energy may be written:

Eelectronic=∫

Vion(r)ρ(r)d3r+Eelectrostatic+ Exc[ρ]

(3)

Therefore, by varying the electron density of the sys-tem through a search of single particle density spaceuntil the minimum energy configuration is found, anexact ground state energy is achieved, and that elec-tron density is the exact ground state energy for thesystem; all other ground state properties are all func-tions of this ground state electron density. However,the exact form of the functionalExc[ρ] is not known,and is approximated by a local function of the density.This local density approximation (LDA) defines theexchange-correlation potential as a function of elec-tron density at a given co-ordinate position (Kohn andSham, 1965). Sometimes it is a better approximation

to use the generalised gradient approximation (GGA)(e.g. Wang and Perdew, 1991), which has a similarform for the exchange-correlation potential, but has itas a function of both the local electron density and themagnitude of its gradient. For simulations of Fe, theuse of GGA is very important, as the use of LDA leadsto the prediction of the wrong ground state (Stixrudeet al., 1998).

The correct description of magnetic effects can bevital in determining the accuracy of the simulation ofthe stability and physical properties of some phases.Thus for example, the body centred cubic (bcc) poly-morph of Fe is only stable at ambient conditions be-cause of its strongly developed ferromagnetism (e.g.see Stixrude and Brown, 1998). Steinle-Neumannet al. (1999)have shown how the calculated equationof state of hcp Fe is dependent upon the correct in-clusion of local spin polarisation, and more recentlyMukherjee and Cohen (2002)have shown the im-portance of using fully unconstrained non-collinearmagnetism to describe the properties of magneti-cally more complex phases. Fortunately, for Fe undercore conditions,Söderlind et al. (1996)have shownthat the high pressures destroy most of the magneticstates that Fe polymorphs exhibit at lower pressure,but nevertheless, studies of highP/T behaviour ofFe must recognise the potential importance of thatmagnetic interactions may have on the physical andthermodynamic properties of the Fe polymorphs.

In any implementation of DFT the descriptionof electron spin states naturally requires a way ofefficiently describing the electron wave functions,Ψi(r). One approach is to describeΨi(r) as a sum ofbasis-sets made up of atomic wave functions,ϕα(r):

Ψi(r) =∑α

ciαϕα(r) (4)

The other approach is to use basis functions that de-scribe the itinerant nature of electrons, and whichare based on the wave function of a free electron,exp(ik·r), wherek is the wave vector of the de Brogliewave. ThusΨi(r) can now be expressed as a sum ofpossible plane-wave basis functions:

Ψi(r) =∑k

cik exp(ik · r) (5)

In fact, both approaches are valid, but in practicefor the large numbers of atoms usually involved in


condensed matter simulations, the plane-wave methodhas to date been much more successful. This might ap-pear at first sight to be surprising, because the electronsin condensed matter are clearly not like free particles,as in most atoms there are tightly bound core electronsconfined around the nucleus. To describe these coreelectrons requires the inclusion of a very large numberof plane-waves. There are techniques, such as the lin-earized augmented plane-wave (LAPW) method, thatdo this and provide very accurate descriptions of thepotential and charge density within the entire crys-tal (for more details see for exampleStixrude et al.,1998). The number of electrons (and hence atoms) towhich this approach can be applied is limited, how-ever, by current computing capacity, and it cannot beused efficiently for large or complex systems.

An alternative to the ‘all electron technique’ is touse electronic wave functions that are expanded ina plane-wave basis-set, with the electron–ion inter-actions described by means of pseudopotentials. Apseudopotential is a modified form of the true po-tential experienced by the electrons (Heine, 1970;Cohen and Heine, 1970; Heine and Weaire, 1970).When they are near the nucleus, the electrons feel astrong attractive potential and this gives them a highkinetic energy. But this means that their de Brogliewavelength is very small, and their wave vector isvery large. As mentioned above, because of this, aplane-wave basis would have to contain so manywave vectors that the calculations would become verydemanding. A remarkable way of eliminating thisproblem and broadening the range of systems thatcan be studied was developed by Heine, Cohen andothers, who showed that it is possible to represent theinteraction of the valence electrons with the atomiccores by a weak effective ‘pseudopotential’ and stillend up with a correct description of the electron statesand the energy of the system. In this way of doingthings, the core electrons are assumed to be in exactlythe same states that they occupy in the isolated atom,which is usually valid. There are several ways of con-structing pseudopotentials, and it is essential to showthat they correctly describe the system under study,but when this is done, it has been found that resultsobtained employing pseudopotentials are as reliableas those from all electron calculations.

More recently the projector augmented wave (PAW)method has increasing been used (e.g.Kresse and

Joubert, 1999). The PAW approach is an all elec-tron method in the sense that it works with the trueKohn–Sham orbitals, rather than orbitals that havebeen “pseudised” in the core regions, and it has thesame level of rigour as other all electron methodssuch as FLAPW. At the same time it is very closelyrelated to the ultra soft pseudopotential technique(Vanderbilt, 1990), and reduces to this if certainwell-defined approximations are used, as shown byKresse and Joubert (1999). The PAW method has,therefore, the computational efficiency of the pseu-dopotential method, but without the minor draw backof the approximation involved in their formulation.

In conclusion, therefore, plane-waves have provedto be very successful for many reasons. The wave func-tions can be made as accurate as necessary by increas-ing the number of plane-waves, so that the method issystematically improvable. Plane-waves are simple sothat the programming is easy, and it also turns out thatthe forces on the ions are straightforward to calculate,so that it is easy to move them. Finally, plane-wavesare unbiased. The calculations are unaffected by theprejudices of the user—an important advantage forany method that is going to be widely used. As suchusing GGA within DFT combined with pseudopoten-tials or PAW methods provides an excellent techniquewith which to accurately explore most crystal struc-tures (although it should be noted that DFT is not en-tirely free of limitations, some of which are discussedfor example inStixrude et al. (1998)). Below, we out-line the application of these methods to the study ofFe. However to study Fe under core conditions, weneed not only to explore the energetics of bonding, butwe are also concerned with the effect of temperatureon the system. This requires us to calculate the Gibbsfree energy of the systems, which can be done eitherusing lattice dynamic or molecular dynamic methods.

3. Lattice dynamics

Lattice dynamics is a semi-classical approach thatuses the quasiharmonic approximation (QHA) todescribe a cell in terms of independent quantisedharmonic oscillators, the frequencies of which varywith cell volume, thus allowing for a descriptionof thermal expansion (e.g.Born and Huang, 1954).The motions of the individual particles are treated


collectively as lattice vibrations or phonons, and thephonon frequencies,ω(q), are obtained by solving:

mω2(q)ei(q) = D(q)ej(q) (6)

wherem is the mass of the atom, and the dynamicalmatrix, D(q), is defined by:

D(q) =∑

ij

(∂2U

∂ui∂uj

)exp(iq · rij) (7)

whererij is the interatomic separation, andui andujare the atomic displacements from their equilibriumposition. For a unit cell containingN atoms, there are3N eigenvalue solutions (ω2(q)) for a given wave vec-tor q. There are also 3N sets of eigenvectors (ex(q),ey(q), ez(q)) which describe the pattern of atomicdisplacements for each normal mode.

The vibrational frequencies of a lattice can be cal-culated ab initio, by standard methods such as thesmall-displacement method (e.g.Kresse et al., 1995).Having calculated the vibrational frequencies, a num-ber of thermodynamic properties may be calculatedusing standard statistical mechanical relations, whichare direct functions of these vibrational frequencies.Thus for example the Helmholtz free energy is givenby:

F = kBT

M∑i

(xi2

+ ln(1 − e−xi ))

(8)

wherexi = hωi/kBT, and the sum is over all theM =3N normal modes. Modelling the effect of pressure isessential if one is to obtain accurate predictions of phe-nomena such as phase transformations and anisotropiccompression. This problem is now routinely beingsolved using codes that allow constant stress, variablegeometry cells in both static and dynamic simulations.In the case of lattice dynamics, the mechanical pres-sure is calculated from strain derivatives, whilst thethermal kinetic pressure is calculated from phonon fre-quencies (e.g.Parker and Price, 1989). The balance ofthese forces can be used to determine the variation ofcell size as a function of pressure and temperature.

The quasi-harmonic approximation assumes that thelattice vibrational modes are independent. Howeverat high temperatures, where vibrational amplitudesbecome large, phonon–phonon scattering becomesimportant, and the QHA breaks down. At ambient

pressure, the QHA is only valid forT < θD, the Debyetemperature, so since we are interested in the extremeconditions of the interior Earth, we would expect tohave to modify this methodology to enable higher tem-peratures to be simulated (see e.g.Ball, 1989), alter-natively we could use molecular dynamics techniques.Matsui et al. (1994)have shown, however, that the in-herent anharmonicity associated with lattice dynamicsdecreases with increasing pressure, and the two tech-niques give very similar results for very high pressureand temperature simulations. It should be noted thatanother approach has been used by some workers(e.g.Stixrude et al., 1997) to calculate the effect ofTon the properties of the highP phases of Fe. They usea mean-field approach known as the particle-in-cell(PIC) method. This method attempts to account foranharmonicity and is computationally efficient. A de-tailed comparison of the relative precision of the PICmethod and molecular dynamic simulations of Fehave been presented byAlfè et al. (2001). Generallythe molecular dynamics approach should be favouredas it involves far fewer approximations, but it is muchmore computationally intensive. We note, however,that the PIC method has the same level of complexityas the QHA, which treats harmonic vibrations exactly,but we have recently shown that the anharmonicitypredicted by the PIC method has thewrong sign(Gannarelli et al., 2003), which weakens the case forpreferring this approximation to that of the QHA.

4. Molecular dynamics

Molecular dynamics is routinely used for mediumto high temperature simulations of minerals and in allsimulations of liquids, where lattice dynamics is ofcourse inapplicable. The method is essentially clas-sical, and its details are presented in, for example,Allen and Tildesley (1987). The interactions betweenthe atoms within the system have traditionally beendescribed in terms of the interatomic potential modelsmentioned earlier, but instead of treating the atomicmotions in terms of lattice vibrations, each ion istreated individually. As the system evolves, the re-quired dynamic properties are calculated iteratively atthe specified pressure and temperature. The ions areinitially assigned positions and velocities within thesimulation box; their co-ordinates are usually chosen


to be at the crystallographically determined sites,whilst their velocities are equilibrated such that theyconcur with the required system temperature, andsuch that both energy and momentum is conserved. Inorder to calculate subsequent positions and velocities,the forces acting on any individual ion are then calcu-lated from the first derivative of the potential function,and the new position and velocity of each ion maybe calculated at each time-step by solving Newton’sequation of motion. Both the particle positions andthe volume of the system, or simulation box, can beused as dynamical variables, as is described in detailin Parrinello and Rahman (1980). The kinetic energy,and therefore temperature, is obtained directly fromthe velocities of the individual particles. With this ex-plicit particle motion, the anharmonicity is implicitlyaccounted for at high temperatures.

Because of advances in computer power, it is nowpossible to perform ab initio molecular dynamics(AIMD), with the forces calculated fully quantummechanically (within the GGA and the pseudopoten-tial or PAW approximations) instead of relying uponthe use of interatomic potentials. The first pioneeringwork in AIMD was that ofCar and Parrinello (1985),who proposed a unified scheme to calculate ab initioforces on the ions and keep the electrons close to theBorn-Oppenheimer surface while the atoms move.We have used in the work summarised below analternative approach, in which the dynamics are per-formed by explicitly minimising the electronic freeenergy functional at each time step. This minimisa-tion is more expensive than a single Car-Parrinellostep, but the cost of the step is compensated by thepossibility of making longer time steps. The molec-ular dynamics simulations presented here have beenperformed using Vienna ab initio simulation package(VASP). In VASP the electronic ground state is cal-culated exactly (within a self-consistent threshold) ateach MD step, using an efficient iterative matrix di-agonalization scheme and the mixer scheme ofPulay(1980). We have also implemented a scheme to ex-trapolate the electronic charge density from one stepto the next, with an efficiency improvement of abouta factor of two (Alfè, 1999). Since we are interestedin finite-temperature simulations, the electronic levelsare occupied according to the Fermi statistics corre-sponding to the temperature of the simulation. Thisprescription also avoids problems with level crossing

during the self-consistent cycles. For more details ofthe VASP code seeKresse and Furthermüller (1996).

It is not possible to obtain free energies directlyfrom MD techniques. These can, however, be obtainedby ‘thermodynamic integration’, which yields the dif-ference between the free energy ($F) of the ab initiosystem and that of a reference system. The basis of thetechnique (see for examplede Wijs et al., 1998) is that$F is the work done on reversibly and isothermallyswitching from the reference total-energy function,Uref, to the ab initio total-energy,U. This switchingis done by passing through intermediate total-energyfunctionsUλ given byUλ = (1− λ)Uref + λU. It is astandard result that the work done is:

$F =∫ 1

0dλ〈U − Uref〉λ (9)

where the thermal average〈U−Uref〉λ is evaluated forthe system governed byUλ. The practical feasibilityof calculating ab initio free energies of liquids and an-harmonic solids depends on finding a reference systemfor which Fref is readily calculable and the difference(U − Uref) is very small. However, we stress that thefinal result is independent on the choice of the refer-ence system. In our studies of liquid Fe, the primaryreference state chosen was an inverse power potential.The full technical details involved in our calculationsare given inAlfè et al. (2002a,b,c). Below, we illus-trate our use of all of these methods in the study of Feat extreme pressure and temperature.

5. The structure of Fe under core conditions

Under ambient conditions, Fe adopts a body centredcubic (bcc) structure, that transforms with temperatureto a face centred cubic (fcc) form, and with pressuretransforms to a hexagonal close-packed (hcp) phase,ε-Fe. The highP/T phase diagram of pure iron itselfhowever is still controversial (seeFig. 1 and also thediscussion inStixrude and Brown (1998)). Various di-amond anvil cell based studies have been interpretedas showing that hcp Fe transforms at high tempera-tures to a phase which has variously been describedas having a double hexagonal close-packed structure(dhcp) (Saxena et al., 1996) or an orthorhombicly dis-torted hcp structure (Andrault et al., 1997). Further-more, high pressure shock experiments have also been


100 200 300

2000

4000

6000

Temp(K)

Pressure (GPa)

Liquid

FCC

BCC

HCP

DHCP?

BCC?

Fig. 1. A hypothetical phase diagram for Fe, incorporating all the experimentally suggested highP/T phase transformations. Our calculationssuggest that the phase diagram is in fact much more simple than this, with hcp Fe being the only highP/T phase stable at core pressures.

interpreted as showing a high pressure solid–solidphase transformation (Brown and McQueen, 1986;Brown, 2001), which has been suggested could be dueto the development of a bcc phase (Matsui and An-derson, 1997). Other experimentalists, however, havefailed to detect such a post-hcp phase (e.g.Shen et al.,1998; Nguyen and Holmes, 1999), and have suggestedthat the previous observations were due either to minorimpurities or to metastable strain-induced behaviour.

Further progress in interpreting the nature and evo-lution of the core would be severely hindered if the un-certainty concerning the crystal structure of the core’smajor chemical component remained unresolved.Such uncertainties can be resolved, however, usingab initio calculations, which we have shown providean accurate means of calculating the thermoelasticproperties of materials at highP andT (e.g.Vocadloet al., 1999). Thermodynamic calculations on hcp Feand fcc Fe at highP/T were reported byWassermanet al. (1996)and byStixrude et al. (1997). They usedab initio calculations to parameterise a tight-bindingmodel; the thermal properties of this model werethen obtained using the particle-in-a-cell method. Thecalculations that we performed (Vocadlo et al., 1999)to determine the highP/T structure of Fe, were the

first in which fully ab initio, non-parameterised meth-ods were used, in conjunction with quasiharmoniclattice dynamics, to obtain free energies under coreconditions of all the proposed candidate Fe structures.

Spin polarised simulations were initially performedon candidate phases (including a variety of distortedbcc and hcp structures and the dhcp phase) at pres-sures ranging from 325 to 360 GPa. These revealed,in agreement withSöderlind et al. (1996), that underthese conditions only bcc Fe has a residual magneticmoment and all other phases have zero magnetic mo-ments. It should be noted however, that the magneticmoment of bcc Fe disappears when simulations areperformed at core pressures and an electronic tem-perature of >1000 K, indicating that even bcc Fe willhave no magnetic stabilisation energy under core con-ditions. We found that at these pressures, both the bccand the suggested orthorhombic polymorph of iron(Andrault et al., 1997) are mechanically unstable. Thebcc phase can be continuously transformed to the fccphase (confirming the findings ofStixrude and Cohen,1995), while the orthorhombic phase spontaneouslytransforms to the hcp phase, when allowed to relax toa state of isotropic stress. In contrast, hcp, dhcp andfcc Fe remain mechanically stable at core pressures,


Fig. 2. The phonon-dispersion curves along a&H&N path in re-ciprocal space for bcc Fe. The solid lines are from our calcula-tions, and the open squares are experimental points reported inGao et al. (1993).

and we were therefore able to calculate their phononfrequencies and free energies.

Although no experimentally determined phonon-dispersion curves exist for hcp Fe, the quality of ourcalculations can be gauged by comparing the cal-culated phonon-dispersion for bcc Fe (done usingfully spin polarised calculations) at ambient pressure,with the existing experimental data.Fig. 2 shows thephonon-dispersion curve for magnetic bcc Fe at ambi-ent conditions compared with data obtained from in-elastic neutron scattering experiments (seeGao et al.,1993); the calculated frequencies are in excellentagreement with the experimental values. The calcu-lated elastic constants for bcc Fe (related to the slopeof the acoustic branches of the phonon-dispersioncurves) are also in a good agreement with experimen-tally determined values (Vocadlo et al., 1997). Morerecently,Mao et al. (2001)measured the phonon den-sity of states of Fe up to 153 GPa using nuclear reso-nant inelastic X-ray scattering. Our calculated phonondensity of states for bcc Fe at ambient pressure andat 3 GPa reported in that paper, are in outstandingagreement with experiment. The agreement with thehcp phase is also very good, but less exact than forthe bcc phase, as our hcp simulations were done ne-glecting magnetic effects. However, nuclear resonantinelastic X-ray scattering gives a very indirect mea-surement of the phonon density of states, and moreresearch in needed to establish the precision of suchhigh pressure measurements.

The thermal pressure of hcp Fe at core conditionshas been estimated to be 58 GPa (Anderson, 1995) and50 GPa (Stixrude et al., 1997); these are in excellentagreement with our calculated thermal pressure for thehcp structure (58 GPa at 6000 K). By analysing thetotal pressure as a function of temperature obtainedfrom our calculations for the potential phases of Fe,we were able to ascertain the temperature as a functionof volume at two pressures (P = 325 GPa andP =360 GPa) that span the inner-core range of pressures.From this we could determine the Gibbs free energy ofthese structures at thoseT andP. We found that, on thebasis of lattice dynamic calculations over the wholeP/T space investigated, the hcp phase of Fe has thelowest Gibbs free energy, and is therefore the stableform of Fe under core conditions.

Despite the fact that bcc Fe is mechanically unstableat high pressure and low temperatures (and so notamenable to lattice dynamical modelling), it has beensuggested that it may become entropically stabilised athigh temperatures (e.g.Matsui and Anderson, 1997).This behaviour is found in some other transition metalphases, such as Ti and Zr (see for exampleLiu andBassett, 1986). We have recently performed moleculardynamic simulations of bcc Fe and have found that thisstructure indeed seems to become mechanically stableabove∼1000 K at core densities. The full details ofits behaviour are still to be resolved, and specificallythe highP/T stability of bcc Fe relative to hcp Fe isthe subject of continuing research.

6. The high P melting of Fe

Having shown how ab initio calculations are be-ing used to establish the sub-solidus phase relations inhigh P Fe, we turn now to completing the descriptionof the highP/T phase diagram of Fe, by considering itsmelting behaviour. An accurate knowledge of the melt-ing properties of Fe is particularly important, as thetemperature distribution in the core is relatively uncer-tain and a reliable estimate of the melting temperatureof Fe at the pressure of the inner-core boundary (ICB)would put a much-needed constraint on core temper-atures. As with the sub-solidus behaviour of Fe, thereis much controversy over its highP melting behaviour(e.g. seeShen and Heinze, 1998). Static compressionmeasurements of the melting temperature,Tm, with the


DAC have been made up to∼200 GPa (e.g.Boehler,1993), but even at lower pressures results forTm dis-agree by several hundred Kelvin. Shock experimentsare at present the only available method to determinemelting at higher pressures, but their interpretation isnot simple, and there is a scatter of at least 2000 K inthe reportedTm of Fe at ICB pressures (seeNguyenand Holmes, 1999).

Since both our calculations and recent experiments(Shen et al., 1998) suggest that Fe melts from theε-phase in the pressure range immediately above60 GPa, we focus here on equilibrium between hcpFe and liquid phases. The condition for two phasesto be in thermal equilibrium at a given temperature,T, and pressure,P, is that their Gibbs free energies,G(P, T), are equal. To determineTm at any pressure,we calculateG for the solid and liquid phases as afunction of T and determine where they are equal.In fact, we calculate the Helmholtz free energy,F(V,T), as a function of volume,V, and hence obtain thepressure through the relationP = −(∂F/∂V)T andGthrough its definitionG = F + PV.

Fig. 3. Our calculated high pressure melting curve for Al (seeVocadlo and Alfe, 2002) is shown passing through a variety of recent highP experimental points.

To obtain melting properties with useful accuracy,free energies must be calculated with high precision,because the free energy curves for liquid and solidcross at a shallow angle. It can readily be shown thatto obtain Tm with a technical precision of 100 K,non-cancelling errors inG must be reduced below10 meV. Errors in the rigid-lattice free energy due tobasis-set incompleteness and Brillouin-zone samplingare readily reduced to a few meV per atom. In thisstudy, the lattice vibrational frequencies were ob-tained by diagonalizing the force-constant matrix; thismatrix was calculated by our (Alfè, 1998) implemen-tation of the small-displacement method described byKresse et al. (1995). The difficulty in calculating theharmonic free energy is that frequencies must be accu-rately converged over the whole Brillouin-zone. Thisrequires that the free energy is fully converged withrespect to the range of the force-constant matrix. Toattain the necessary precision we used repeating cellscontaining 36 atoms, and to show that such a systemgives converged energies, we performed some highlycomputationally demanding calculations on cells of


up to 150 atoms. The anharmonic contributions to thefree energy of the solid, and the free energies of theliquid were calculated by molecular dynamics, usingthermodynamic integration.

To confirm that the methodology can be used ac-curately to calculate melting temperatures, we mod-elled the well studied highP melting behaviour of Al(de Wijs et al., 1998; Vocadlo and Alfè, 2002). Fig. 3shows the excellent agreement that we obtained forthis system. In 1999 we published an ab initio melt-ing curve for Fe (Alfè et al., 1999). Since the workreported in that paper, we have improved our descrip-tion of the ab initio free energy of the solid, and haverevised our estimate ofTm of Fe at ICB pressures tobe∼6250 (seeFig. 4 andAlfè et al. (2002b,d)), withan error of±300 K. A full analysis of the errors hasbeen reported inAlfè et al. (2002b,d). For pressuresP < 200 GPa (the range covered by DAC experiments)our curve lies∼900 K above the values ofBoehler

Fig. 4. Our calculated highP melting curve of Fe (plotted as a solid black line) is shown passing through the shock-wave datum (openblue circle) ofBrown and McQueen (1986). Other data shown includes: our melting curve corrected for the GGA pressure error (blackdashed line), Belonosko’s melting curve (blue line), Belonosko’s melting data corrected for potential errors (black dots), Laio et al.’smelting curve (green line), Boehler’s DAC curve (green dashed line), Shen’s data (green diamonds), Yoo’s shock data (open squares),William’s melting curve (pale green dashed line). (For interpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

(1993)and∼200 K above the more recent values ofShen et al. (1998)(who stress that their values are onlya lower bound toTm). Our curve falls significantlybelow the shock-based estimates for theTm of Yooet al. (1993), in whose experiments temperature wasdeduced by measuring optical emission (however, thedifficulties of obtaining temperature by this method inshock experiments are well known), but accords al-most exactly with the shock data value ofBrown andMcQueen (1986)and the new data ofNguyen andHolmes (1999). For melting at ICB pressure, we cal-culate$Vm/V = 1.8%,$Sm = 1.05R J K−1 mol−1,and so obtain a latent heat of fusion of 55 kJ mol−1.

There are other ways of determining the meltingtemperature of a system by ab initio methods, includ-ing performing simulations that model co-existingliquid and crystal phases. The melting temperature ofsuch a system can then be inferred by seeing which ofthe two phases grows during the course of a series of


simulations at different temperatures. This approachhas been used byLaio et al. (2000)and byBelonoskoet al. (2000)to study the melting of Fe. In their studiesthey modelled Fe melting using interatomic potentialsfitted to ab initio surfaces. We discovered, however,that their fitted potentials did not simultaneously de-scribe the energy of the liquid and crystalline phaseswith the same precision, and so their simulations donot represent the true melting behaviour of Fe, butrather that of the fitted potential. We have recentlyalso used the co-existence method (Alfè et al., 2002d),but with a model potential fitted to our own ab ini-tio calculations. Initially, our raw model failed to givethe same melting temperature as obtained from our abinitio free energy method, but when the results werecorrected for the free energy mismatch of the modelpotential with respect to the ab initio energies of liquidand solid, the results for the two methods came intoagreement. Thus it would seem as a general principlethat there is a way to correct for the shortcomings ofmodel potential co-existence calculations, namely onemust calculate the free energy differences between themodel and the ab initio system for both the liquid andsolid phases. This difference in free energy betweenliquid and solid can then be transformed into an ef-fective temperature correction. When this is done toBelonosko’s data, there is excellent agreement withour ab initio melting curve for Fe.

As recently highlighted byCahn (2001), there isstill scope for further work on the difficult problem ofthe modelling of melting, but for the highP meltingof Fe it appears that there may be more problems withreconciling divergent experimental data than there arein obtaining accurate predictions ofTm from ab initiostudies.

7. The thermodynamic properties of hcp Fe

In the course of performing the free energy calcula-tions outlined above, we calculated a number of otherthermodynamic properties of Fe, which are of geo-physical significance. In this section, we outline ourcalculated thermodynamic properties of hcp Fe, andin the following section, we outline the properties ofliquid Fe. As the only experimental data on Fe un-der true coreP andT comes from shock experiments,we present first the results of our ab initio studies of

hcp Fe along the Hugoniot. A full technical discus-sion of the calculations summarised here is presentedin Alfè et al. (2001). However, we note that all thecalculated thermodynamic properties were derived byinitially determining the Helmholtz free energy, thevolume dependence of which was used to infer thepressure and hence the Gibbs free energy. The calcu-lated energies were fitted throughout to a third orderBirch-Murnaghan equation of state or a polynomialfunction.

In a shock experiment, conservation of mass, mo-mentum and energy requires that the pressure,PH,the molar internal energyEH, and the molar vol-ume VH in the compression wave are related by theRankine–Hugoniot formula:12PH(V0 − VH) = EH − E0 (10)

whereE0 andV0 are the internal energy and volume inthe zero-pressure state before the arrival of the wave.The quantities directly measured are the shock-waveand material velocities, which allow the values ofPHandVH to be deduced. From a series of experiments,PH as a function ofVH (the so-called Hugoniot) canbe derived. The measurement of temperature in shockexperiments has been attempted but is problematic(e.g. Yoo et al., 1993). The Hugoniot curvePH(VH)is straightforward to compute from our results: fora givenVH, one seeks the temperature at which theRankine–Hugoniot relation is satisfied; from this, oneobtainsPH (and, if required,EH). In experiments onFe, V0 and E0 refer to the zero-pressure bcc crystal,and we obtain their values directly from GGA calcu-lations.

Our ab initio Hugoniot is in good agreement withthat measured byBrown and McQueen (1986), withdiscrepancies ranging from 10 GPa atV = 7.8 Å3 to12 GPa atV = 8.6 Å3. These discrepancies can beregarded as giving an indication of the intrinsic ac-curacy of the GGA itself. Another way of lookingat the accuracy to be expected of the GGA is to re-calculate the Hugoniot using the experimental valueof the bccV0 (11.8 Å3, compared with the ab initiovalue of 11.55 Å3). A similar comparison with the ex-perimental Hugoniot was given in the tight-bindingtotal-energy work ofWasserman et al. (1996)and theiragreement was as good as ours.

Our Hugoniot temperature as a function of pressureis compared with the experimental results ofYoo et al.


Fig. 5. Experimental and ab initio temperature as a function ofpressure on the Hugoniot. The black circles are the data ofYooet al. (1993)and the white diamonds are those ofBrown andMcQueen (1986). The solid curve is the ab initio data ob-tained when the calculated volume of bcc Fe is used in theHugoniot-Rankine equation; the dotted curve is the same, but withthe experimental equilibrium volume of bcc Fe. The comparisonis meaningful only up to a pressure of∼250 GPa, at which pointthe experiments indicate melting.

(1993) in Fig. 5. We also include in the figure theestimates for Hugoniot temperature due toBrown andMcQueen (1986)The latter estimates were based onthe basic thermodynamic relation:

dT = −T( γV

)dV + [(V0 − V)dP + (P − P0)dV ]

2Cv

(11)

between infinitesimal changes of dT, dV, and dPalong the Hugoniot. This relation contains theconstant-volume specific heatCv and the Grüneisenparameterγ, for which Brown and McQueen had tomake assumptions. Our ab initio temperatures fallsubstantially below those of Yoo et al., and this sup-ports the suggestion ofWasserman et al. (1996)thatthe Yoo et al. measurements overestimate the Hugo-niot temperature by∼1000 K. On the other hand, ourtemperatures agree rather closely with the Brown andMcQueen estimates. When we examine below theirassumptions aboutCv and γ, we shall see that theywere reasonable, though the agreement between theirtemperatures and ours is also partly due to cancel-lation of errors between terms they use to evaluateEq. (11).

Fig. 6. Experimental and ab initio adiabatic bulk modulus (KS )on the Hugoniot. The white diamonds are the data ofJeanloz(1979)and the crosses are those ofBrown and McQueen (1986).The solid curve is the ab initio data obtained when the calculatedvolume of bcc Fe is used in the Hugoniot-Rankine equation; thedotted curve is the same, but with the experimental equilibriumvolume of bcc Fe.

A further quantity that can be extracted from shockexperiments is the bulk sound velocityvB as a func-tion of atomic volume on the Hugoniot, which is givenby vB = (KS/ρ)

1/2, whereKS≡ − V(dP/dV)S is theadiabatic bulk modulus andρ is the mass density.SinceKS can be calculated from our ab initio pressureand entropy as functions ofV and T, our calculatedKS can be directly compared with experimental val-ues (Fig. 6). Here, there is a greater discrepancy thanone would wish, with the theoretical values fallingsignificantly above theKS values of bothBrown andMcQueen (1986)and Jeanloz (1979), although wenote that the two sets of experimental results disagreeby an amount comparable with the discrepancy be-tween theory and experiment.

We turn now to the more general thermodynamicproperties of hcp Fe off the Hugoniot, and make somefurther comparisons with the predictions ofStixrudeet al. (1997)andWasserman et al. (1996). Our resultsare presented as a function of pressure on isothermsat T = 2000, 4000, and 6000 K. At each tempera-ture, we give results only for the pressure range where,according to our calculations the hcp phase is ther-modynamically stable. In comparing with the predic-tions of Stixrude et al. (1997)and Wasserman et al.(1996), we show the explicit numerical results fromWasserman et al. (1996)for thermodynamic quantities


Fig. 7. Total constant-volume specific heat per atom (Cv, in units ofkB) of hcp Fe as a function of pressure on isothermsT = 2000 K(continuous curves), 4000 K (dashed curves), and 6000 K (dottedcurves). Heavy and light curves show present results and those ofStixrude et al. (1997), respectively.

on the 2000 K isotherm. For the higher temperatures,we rely on the approximate parameterised formulasgiven inStixrude et al. (1997).

The total constant-volume specific heat per atomCv

(Fig. 7) emphasises again the importance of electronicexcitations. In a purely harmonic system,Cv would beequal to 3kB, and it is striking thatCv is considerablygreater than that even at the modest temperature of2000 K, while at 6000 K it is nearly doubled. The de-crease ofCv with increasing pressure evident inFig. 7comes from the suppression of electronic excitationsby high compression, and to a smaller extent from thesuppression of anharmonicity. We note that ourCv

values are significantly higher than those ofStixrudeet al. (1997)and Wasserman et al. (1996); the mainreason for this seems to be our inclusion of anhar-monic corrections via AIMD and theT-dependenceof harmonic frequencies. As stated above,Brown andMcQueen (1986)made assumptions about the highP/T behaviour ofCv in order to estimate the Hugoniottemperature. Their assumptions were that the latticecontribution toCv is equal to 3kB above the Debyetemperature and that the electronic contribution couldbe represented in the formβe(V/Vref)

δT , whereVref isa reference density, andβe andδ are constants whosevalues were taken from earlier theoretical calcula-tions. Since anharmonic and electronic contributionsare negligible at low temperatures, our calculatedCv

Fig. 8. The thermal expansivity (α) of hcp Fe as a function ofpressure on isothermsT = 2000 K (continuous curves), 4000 K(dashed curves), and 6000 K (dotted curves). Heavy and lightcurves show present results and those ofStixrude et al. (1997),respectively. The black circle is the experimental value of Duffyand Ahrens (1993) atT = 5200±500 K. Diamonds are data fromBoehler et al. (1990)for temperatures between 1500 and 2000 K.

agrees with the Brown and McQueen values on thelow P/T part of the Hugoniot. However, ourCv risesslightly faster, mainly because of anharmonicity, andbecomes∼3% higher than theirs at 200 GPa, the dif-ference between the two decreasing again thereafter.

The thermal expansivityα (Fig. 8) is one of the fewcases where we can compare with DAC measurements(e.g.Boehler et al., 1990). The latter show thatα de-creases strongly with increasing pressure and our abinitio results fully confirm this. Our results also showthat α increases significantly with temperature. Bothtrends are also shown by the calculations ofStixrudeet al. (1997)andWasserman et al. (1996), though thelatter differ from ours in showing considerably largervalues of a at low pressure and temperature. The prod-uct αKT of expansivity and isothermal bulk modu-lus, which is equal to (dP/dT)v, is important becauseit is sometimes assumed to be independent of pres-sure and temperature over a wide range of conditions,and this constancy is used to extrapolate experimentaldata. Our predicted isotherms forαKT (Fig. 9) indi-cate that its dependence onP is indeed weak, espe-cially at low temperatures, but that its dependence onT certainly cannot be ignored, since it increases byat least 30% asT goes from 2000 to 6000 K at high


Fig. 9. The product of the expansion coefficient (α) and the isother-mal bulk modulus (KT) as a function of pressure on isothermsT = 2000 K (continuous curves), 4000 K (dashed curves), and6000 K (dotted curves). Heavy and light curves show present re-sults and those ofStixrude et al. (1997), respectively.

pressures.Wasserman et al. (1996)come to qualita-tively similar conclusions, and they also find valuesof ∼10 MPa K−1 at T = 2000 K. However, we notethat the general tendency in our results forαKT to in-crease with pressure at low pressures is not found inthe results ofWasserman et al. (1996)at 2000 K. Inparticular, they found a marked increase ofαKT withdecreasingP, which does not occur in our results.

The thermodynamic Grüneisen parameterγ =V(dP/dE)v = αKTV/Cv plays an important role inhigh pressure physics, because it relates the thermalpressure and the thermal energy. Assumptions aboutthe value ofγ are frequently used in reducing shockdata from the Hugoniot to an isothermal state. If oneassumes thatγ depends only onV, then the thermalpressure and energy are related by:

PthV = γEth (12)

the well known Mie-Grüneisen equation of state.At low temperatures, where only harmonic phononscontribute toEth and Pth, γ should indeed be tem-perature independent above the Debye tempera-ture, becauseEth = 3kBT per atom, andPthV =−3kBT(dlnω/dlnV) = 3kBTγph, so thatγ = γph (thephonon Grüneisen parameter) which depends only onV. But in highT Fe, the temperature independence ofγ will clearly fail, because of electronic excitations

Fig. 10. The Grüneisen parameter (γ) as a function of pressureon isothermsT = 2000 K (continuous curves), 4000 K (dashedcurves), and 6000 K (dotted curves). Heavy and light curves showpresent results and those ofStixrude et al. (1997), respectively.

and anharmonicity. Our results forγ (Fig. 10) indi-cate that it varies rather little with either pressure ortemperature in the region of interest. At temperaturesbelow∼4000 K, it decreases with increasing pressure,as expected from the behaviour ofγph. This is alsoexpected from the often-used empirical rule of thumbγ = (V/V0)

q, whereV0 is a reference volume andq is a constant exponent usually taken to be roughlyunity. SinceV decreases by a factor of about 0.82 asP goes from 100 to 300 GPa, this empirical relationwould makeγ decrease by the same factor over thisrange, which is roughly what we see. However, thepressure dependence ofγ is very much weakened asT increases, until at 6000 K,γ is almost constant. Ourresults agree moderately well with those ofStixrudeet al. (1997)andWasserman et al. (1996)in giving avalueγ = 1.5 at high pressures, but at low pressuresthere is a significant disagreement, since they find astrong increase ofγ to values of over 2.0 asP > 0,whereas our values never exceed 1.6.

In making their estimates of the Hugoniot tem-perature, Brown and McQueen (1986)assumedthat (dE/dP)v = V /� is a constant equal to2.85 × 10−6 m3 mol−l . This implies thatγ is ∼2.2on the low P/T part of the Hugoniot, whereas ourcalculations giveγ ∼ 1.5. However, with increasingpressure, the Brown–McQueen value ofγ approachesours, being only∼8% higher at 200 GPa. Given the


differences between theirCv andγ values and ours,one might expect a larger disagreement between theirHugoniot temperatures and ours. However, it turnsout that there is some cancellation between the differ-ences in the various terms ofEq. (11)which bringsthe temperature curves into the quite close agreementthat we have seen (Fig. 6).

The elastic constants of hcp Fe at 39 and 211 GPahave been measured in an experiment reported byMao et al. (1999). Calculations of a thermal elasticconstants for hcp Fe have been reported byStixrudeand Cohen (1995), Söderlind et al. (1996)andSteinle-Neumann et al. (1999). These values, togetherwith values that we have calculated are presentedin Table 1, and plotted as a function of density inFig. 11a and b. Although there is some scatter on thereported values ofc12, overall the agreement betweenthe experimental and various ab initio studies are ex-cellent. The resulting bulk and shear moduli and theseismic velocities of hcp Fe as a function of pressure

Table 1A compilation of elastic constants (cij , in GPa), bulk (K) and shear (G) moduli (in GPa), and longitudinal (vP ) and transverse (vS ) soundvelocity (in km s−1) as a function of density (ρ, in g cm−3) and atomic volume (in Å3 per atom)

V ρ c11 c12 c13 c33 c44 c66 K G vP vS

Stixrude and Cohen9.19 10.09 747 301 297 802 215 223 454 224 8.64 4.717.25 12.79 1697 809 757 1799 421 444 1093 449 11.50 5.92

Steinle-Neumann et al.8.88 10.45 930 320 295 1010 260 305 521 296 9.36 5.327.40 12.54 1675 735 645 1835 415 470 1026 471 11.49 6.136.66 13.93 2320 1140 975 2545 500 590 1485 591 12.77 6.51

Mao et al.9.59 9.67 500 275 284 491 235 113 353 160 7.65 4.067.36 12.60 1533 846 835 1544 583 344 1071 442 11.48 5.92

Söderlind et al.9.70 9.56 638 190 218 606 178 224 348 200 8.02 4.577.55 12.29 1510 460 673 1450 414 525 898 448 11.03 6.046.17 15.03 2750 893 1470 2780 767 929 1772 789 13.70 7.24

This study9.17 10.12 672 189 264 796 210 242 397 227 8.32 4.748.67 10.70 815 252 341 926 247 282 492 263 8.87 4.968.07 11.49 1082 382 473 1253 309 350 675 333 9.86 5.387.50 12.37 1406 558 647 1588 381 424 900 407 10.80 5.746.97 13.31 1810 767 857 2007 466 522 1177 500 11.77 6.136.40 14.49 2402 1078 1185 2628 580 662 1592 630 12.95 6.59

In this tableK = (〈c11〉 + 2〈c12〉)/3 andG = (〈c11〉 − 〈c12〉 + 3〈c44〉)/5, where〈c11〉 = (c11 + c22 + c33)/3, etc. Previous calculated valuesare fromStixrude and Cohen (1995), Steinle-Neumann et al. (1999), Söderlind et al. (1996)and this study. The experimental data ofMaoet al. (1999)is also presented.

are shown inFigs. 12 and 13, along with experimentaldata. The calculated values compare well with ex-perimental data at higher pressures, but discrepanciesat lower pressures are probably due to the neglectof magnetic effects in the simulations. The effect oftemperature on the elastic constants of Fe have beenmodelled bySteinle-Neumann et al. (2001)using thePIC method. They found a significant change in thec/a ratio of the hcp structure with temperature, whichcauses a marked reduction inc33, c44 and c66 withincreasing temperature. This led them to concludethat increasing temperature reverses the sense of thesingle crystal longitudinal anisotropy of hcp Fe. Ourcalculations (Gannarelli et al., 2003) have failed thusfar to reproduce the strong effect of temperature onc/a seen by Steinle-Neumann et al., and if this resultis confirmed by more precise molecular dynamic sim-ulations, then it will have important implications forthe interpretation of the seismic tomography of theinner-core.


0

1000

2000

3000

9 11 13 1

Density (g/cc)

cij(G

Pa)

5

9 11 13 1

Density (g/cc)

5

0

1000

2000

3000

cij (

GP

a)

(a)

(b)

Fig. 11. Plot of elastic constants of hcp Fe given inTable 1as a function of density: (a)c11 black diamonds,c12 white squares,c44 blackcircles, (b)c33 white diamonds,c13 black squares andc66 white circles.

8. Rheological and thermodynamics propertiesof liquid Fe

We have reported our investigation into the viscos-ity, diffusion and thermodynamic properties of Fe inthe liquid state inde Wijs et al. (1998)and inAlfè et al.(2000). The technical details of the simulations canbe found in these references. In brief, we performedPAW simulations at the 15 thermodynamic stateslisted in Table 2, all these simulations being done onthe 67-atom system. With these simulations, we coverthe temperature range 3000–8000 K and the pressurerange 60–390 GPa, so that we more than cover therange of interest for the Earth’s liquid core. The table

Table 2Pressure (in GPa) calculated as a function of temperature (T) anddensity for liquid Fe

T (K) � (kg m−3)

9540 10700 11010 12130 13300

3000 604300 132 (135)5000 140 (145)6000 90 151 (155) 170 (170) 251 (240) 360 (335)7000 161 181 264 (250) 375 (350)8000 172 191 275 390 (360)

Experimental estimates are in parenthesis, fromAnderson andAhrens (1994).


0

400

800

1200

1600

2000

0 100 200 300 400

Pressure (GPa)

Mod

ulus

(G

Pa)

Fig. 12. Plot of bulk modulus (diamonds) and shear modulus (squares) for hcp Fe as a function of pressure, with values taken fromTable 1.Black diamonds and squares represent ab initio values, while white diamonds and squares represent values obtained from experimentallydetermined elastic constants.

reports a comparison of the pressures calculated in thesimulations with the pressures deduced byAndersonand Ahrens (1994)from a conflation of experimentaldata, and inFig. 14a–fwe show our calculated values

2

6

10

14

0 100 200 300 400

Pressure (GPa)

Vel

ocity

(km

/s)

Fig. 13. Plot of aggregatevP (diamonds) andvS (squares) wave velocity for hcp Fe as a function of pressure, with values taken fromTable 1.Black diamonds and squares represent ab initio values, while white diamonds and squares represent values obtained from experimentallydetermined elastic constants. White circles are the experimental data of Fiquet et al. (2001).

of density, thermal expansion coefficient, adiabatic andisothermal bulk moduli, heat capacity (Cv), Grüneisenparameter, and bulk sound velocity respectively.These values are in close accord with the estimates


Fig. 14. Panel showing the calculated values of (a) density, (b) thermal expansion coefficient, (c) adiabatic (dashed lines) and isothermalbulk (solid lines) moduli, (d) heat capacity (Cv), (e) Grüneisen parameter, and (f) bulk sound velocity for liquid Fe, all as a function ofPat temperatures between 4000 and 8000 K.


Fig. 14. (Continued ).

available inAnderson and Ahrens (1994), indeed, ourfirst principles pressures reproduce the experimentaldensity values to within 2–3% at low densities, butthey are systematically too high by∼7% at high den-sities. It is not clear yet whether the high-density dis-crepancies indicate a real deficiency in the ab initio

calculations rather than problems in the interpretationof the experimental data.

It is commonly found that increasing temperaturesignificantly reduces the sound velocity in a liquid.So for iron at atmospheric pressure, dvP /dT hasbeen estimated byAnderson and Ahrens (1994)to


Fig. 14. (Continued ).

be ∼−0.54 ± 0.21 ms−1 K−1. These authors subse-quently reviewed the available highP data for liquidiron, and concluded that “the pure Fe isentopes arenearly indistinguishable from one another in sound

velocity”, but they did not directly calculate higherpressure values of dvP /dT. However, the highP be-haviour of dvP /dT is important, as pointed out byGubbins et al. (2003), as it has a direct relationship


with the variation of the Grüneisen parameter alongthe core adiabat, thus:

(∂γ

∂P

)S

= −(

γ

KS

)[2 + γ −

(∂KS

∂P

)S

−(

1

αv2P

)(∂v2

P

∂T

)P

](13)

which in turn affects the adiabatic temperature gradi-ent in the outer core, and hence the effectiveness ofthe geodynamo.

In the absence of any firmer data at the time,Gubbins and Masters (1979)took (∂v2

P/∂T )P to be

Fig. 15. (a) A plot of (∂γ/∂P)S (in GPa−1) againstP (in GPa) showing curves obtained by differentiation of the curves shown in 15b andby evaluation ofEq. (13). (b) The ab initio calculated variation of the Grüneisen parameter of liquid Fe along the core adiabat. (c) Thecalculated variation of (∂v2

P/∂T )P (in m2 s−2 K−1) of liquid iron as a function ofP (in GPa).

−2000 m2 s−2 K−1 (c.f. the ambient pressure valueobtained fromAnderson and Ahrens (1994)of about−4000 m2 s−2 K−1) when they developed their modelof the core and the geodynamo. However, it is nowpossible to determine higher order derivatives such as(∂γ/∂P)S and (∂v2

P/∂T )P from our ab initio calcula-tions. The robustness of the derivation of these higherterms is shown inFig. 15a, which shows (∂γ/∂P)S asobtained from numerical differentiation of the varia-tion of our calculated Grüneisen parameter along theadiabat (Fig. 15b), and that obtained by evaluatingEq. (13), using the values calculated for (∂v2

P/∂T )Pshown inFig. 15c. It is important to note that thesehigh pressure values (∂v2

P/∂T )P are significantly


Table 3The diffusion coefficient (D) and the viscosity (η) from our abinitio simulations of liquid Fe at a range of temperatures anddensities

T (K) � (kg m−3)

9540 10700 11010 12130 13300

D (10−9 m2 s−1)3000 4.0± 0.44300 5.2± 0.25000 7.0± 0.76000 14± 1.4 10± 1 9 ± 0.9 6 ± 0.6 5 ± 0.57000 13± 1.3 11± 1.1 9 ± 0.9 6 ± 0.6

η (mPa s)3000 6± 34300 8.5± 15000 6± 36000 2.5± 2 5 ± 2 7 ± 3 8 ± 3 15 ± 57000 4.5± 2 4 ± 2 8 ± 3 10 ± 3

The error estimates come from statistical uncertainty due to theshort duration of the simulations.

smaller than the ambient pressure value, and that anymodel of core behaviour should use these more ap-propriate values. Although not explicitly presented inAnderson and Ahrens (1994), it is possible to estimatea value for (∂v2

P/∂T )P from theirFig. 10, which yieldsa value of∼−140 m2 s−2 K−1 at the ICB, in excellentagreement with our ab initio calculation.

In Table 3, we report values for the self diffusionof Fe in liquid Fe, and the viscosity of liquid Fe forthe same range ofP andT conditions reported above.Since most of the simulations reported inTable 3arerather short (typically no more than 4 ps) the statisticalaccuracy onD andη is not great (the error estimatesalso reported inTable 3). These ab initio simulationsdemonstrate that under Earth’s core conditions liq-uid Fe is a typical simple liquid. In common withother simple liquids, like liquid Ar and Al, it has aclose-packed structure, the coordination number inthe present case being∼13. In fact, there appears tobe a truly remarkable simplicity in the variation ofthe liquid properties of Fe with thermodynamic state.To show the relative independence of the radial dis-tribution function (rdf) with temperature, we report inFig. 16 the rdf, g(r), for the five temperatures 4300,5000, 6000, 7000, and 8000 K at the same densityρ = 10,700 kg m−3. The effect of varying temper-ature is clearly not dramatic, and consists of theexpected weakening and broadening of the structure

Fig. 16. The variation of the radial distribution function (g(r))of liquid Fe with temperature from ab initio simulations at thefixed density ofρ = 10.7 g cm−3. Results are shown for the fivetemperaturesT = 4300, 5000, 6000, 7000 and 8000 K.

with increasingT. For the entire pressure-temperaturedomain of interest for the Earth’s outer core, thediffusion coefficient,D, and viscosity,η, are compa-rable with those of typical simple liquids, D being∼5×10−9 m2 s−1 andη being in the range 8–15 mPa s,depending on the detailed thermodynamic state. Sim-ilar estimates for viscosity of liquid Fe are presentedby Stixrude et al. (1998), based on calculations usingthe tight-binding approximation.

Since the Earth’s outer core is in a state of convec-tion, the temperature and density will lie on adiabats.It is straightforward to show that the variation ofDandη along adiabats will be rather weak. For example,if we take the data for high pressure liquid iron com-piled byAnderson and Ahrens (1994)then the adiabatfor T = 6000 K at the inner-core boundary pressure330 GPa hasT = 4300 K at the core mantle boundary(CMB) pressure of 135 GPa. Taking the densities atthese two points from the same source, we find the liq-uid structure virtually unchanged, apart from a triviallength scaling. For all practical purposes, then, it canbe assumed that variation of thermodynamic condi-tions across the range found in the core has almost noeffect on the liquid structure. For the same reasons, thediffusion coefficientD and viscosity also show littlevariation, so that the diffusion coefficientD is 5±0.5×10−9 m2 s−1 without significant variation as one goesfrom ICB to CMB pressures along the adiabat, and inparallelη goes from 15± 5 to 8.5 ± 1 mPa s. Thesecalculations finally resolve the issue of the viscosity


of the outer core, which historically was considered tobe uncertain to within 13 orders of magnitude!

9. Conclusion

The past decade has seen a major advance in theapplication of ab initio methods in the solution ofhigh pressure and temperature geophysical problems,thanks to the rapid developments in high performancecomputing. In all of our studies of solid and liquid Fesummarised above, we have made strenuous efforts todemonstrate the robustness and reliability of our cal-culations. We have shown that our results are com-pletely robust with respect to all the main technicalfactors: size of simulated system,k-point sampling,and choice of ab initio method. At present, ab ini-tio dynamical simulations of the type presented hereare only practicable with the PAW or pseudopoten-tial techniques. However, our comparisons leave littledoubt that if such simulations were feasible with otherDFT techniques, such as FLAPW, virtually identicalresults would be obtained. As a result, we are confidentthat we are now in a position to calculate from firstprinciples the free energies of solid and liquid phases,and hence to determine both the phase relations andthe physical properties of planetary forming minerals.

In the future, we look forward to the advent of rou-tinely available ‘terascale’ computing. This will opennew possibilities for geophysical modelling. Thus, wewill be able to model more complex and larger sys-tems, to investigate for example solid state rheologicalproblems, or physical properties such as thermal andelectrical conductivity. However, we recognise that theDFT methods we currently use are still approximate,and fail for example to describe the band structure ofimportant phases such as FeO. In the future we intendto use terascale facilities to implement more demand-ing but more accurate techniques, such as those basedon quantum Monte Carlo methods.

Acknowledgements

We would like to thank Lars Stixrude for hisvery helpful comments on an early version of ourmanuscript. DA and LV are supported by Royal So-ciety University Research Fellowships. MJG thanks

GEC and Daresbury Laboratory for their support. Thiswork was supported by NERC grants GR3/12083 andGR9/03550. The calculations were run on the CrayT3D and Cray T3E machines at Edinburgh and theManchester CSAR Centre and on the Origin 2000machine at the UCL HiPerSPACE Centre.

References

Alfè, D., 1998. Program available fromhttp://www.chianti.geol.ucl.ac.uk/∼dario/phon.tar.z.

Alfè, D., 1999. Ab initio molecular dynamics, a simple algorithmfor charge extrapolation. Comput. Phys. Commun. 118, 31–33.

Alfè, D., Gillan, M.J., Price, G.D., 1999. The melting curve of ironat the pressures of the Earth’s core from ab initio calculations.Nature 401, 462–464.

Alfè, D., Kresse, G., Gillan, M.J., 2000. Structure and dynamicsof liquid Iron under Earth’s core conditions. Phys. Rev. 61,132–142.

Alfè, D., Price, G.D., Gillan, M.J., 2001. Thermodynamics ofhexagonal-close-packed iron under Earth’s core conditions.Phys. Rev. B64, 045123.

Alfè, D., Gillan, M.J., Price, G.D., 2002a. Composition andtemperature of the Earth’s core constrained by combining abinitio calculations and seismic data. Earth Planet. Sci. Lett. 195,91–98.

Alfè, D., Price, G.D., Gillan, M.J., 2002b. Iron under Earth’score conditions: liquid-state thermodynamics and high-pressuremelting curve from ab initio calculations. Phys. Rev.B65 (165118), 1–11.

Alfè, D., Gillan, M.J., Price, G.D., 2002c. Ab initio chemicalpotential of solid and liquid solutions and chemistry of theEarth’s core. J. Chem. Phys. 116, 7127–7136.

Alfè, D., Gillan, M.J., Price, G.D., 2002d. Complementaryapproaches to the ab initio calculation of melting properties. J.Chem. Phys. 116, 6170–6177.

Allègre, C.J., Poirier, J.P., Humler, E., Hofmann, A.W., 1995. Thechemical composition of the Earth. Phys. Earth Planet. Interiors.134, 515–526.

Allen, M.P., Tildesley, D.J., 1987. Computer Simulation of Liquids.Oxford University Press, Oxford, UK.

Anderson, O.L., 1995. Equations of State of Solids for Geophysicsand Ceramic Science. Oxford University Press, Oxford, UK.

Anderson, W.W., Ahrens, T.J., 1994. An equation of state forliquid iron and implication for the Earth’s core. J. Geophys.Res. 99, 4273–4284.

Andrault, D., Fiquet, G., Kunz, M., Visocekas, F., Häusermann,D., 1997. The orthorhombic structure of iron: an in situ studyat high temperature and high pressure. Science 278, 831–834.

Ball, R.D., 1989. Anharmonic lattice dynamics. In: Stoneham,A.M. (Ed.), Anharmonic Lattice Dynamics in Ionic Solids atHigh Temperatures. World Scientific, Singapore.

Belonosko, A.B., Ahuja, R., Johansson, B., 2000. Quasi-ab initiomolecular dynamic study of Fe melting. Phys. Rev. Lett. 84,3638–3641.

http://www.chianti.geol.ucl.ac.uk/~dario/phon.tar.z

http://www.chianti.geol.ucl.ac.uk/~dario/phon.tar.z


Birch, F., 1952. Elasticity and the constitution of the Earth’sinterior. J. Geophys. Res. 57, 227–286.

Boehler, R., 1993. Temperature in the Earth’s core from the meltingpoint measurements of iron at high static pressures. Nature 363,534–536.

Boehler, R., 2000. High-pressure experiments and the phasediagram of lower mantle and core materials. Rev. Geophys. 38,221–245.

Boehler, R., von Bargen, N., Chopelas, A., 1990. Melting, thermalexpansion and phase transitions of iron at high pressure. J.Geophys. Res. 95, 21731–21736.

Born, M., Huang, K., 1954. Dynamical Theory of Crystal Lattices.Oxford University Press, Oxford, UK.

Brown, J.M., 2001. The equation of state of iron to 450 GPa:another high pressure solid 18 phase? Geophys. Res. Lett. 28,4339–4342.

Brown, J.M., McQueen, R.G., 1986. Phase transitions, Grüneisenparameter and elasticity for shocked iron between 77 GPa and400 Gpa. J. Geophys. Res. 91, 7485–7494.

Cahn, R.W., 2001. Melting from within. Nature 413, 582–583.Car, R., Parrinello, M., 1985. Unified approach for molecular

dynamics and density functional theory. Phys. Rev. Lett. 55,2471–2474.

Cohen, M., Heine, V., 1970. The fitting of pseudopotentialsto experimental data and their subsequent application. In:Ehrenreich, H., Seitz, F., Turnbull, D. (Eds.), Solid StatePhysics, vol. 24, pp. 37–248.

de Wijs, G.A., Kresse, G., Gillan, M.J., 1998. First order phasetransitions by first principles free energy calculations: themelting of Al. Phys. Rev. B57, 8233–8234.

Gannarelli, C.M.S., Alfè, D., Gillan, M.J., 2003. The particle-in-cell model for ab initio thermodynamics: implications for theelastic anisotropy of the Earth’s inner core. Phys. Earth Planet.Interiors, submitted for publication.

Gao, F., Johnston, R.L., Murrell, J.N., 1993. Empirical many-bodypotential energy functions for iron. J. Phys. Chem. 97, 12073–12082.

Gillan, M.J., 1997. The virtual matter laboratory. Contemp. Phys.38, 115–130.

Heine, V., 1970. The pseudopotential concept. In: Ehrenreich, H.,Seitz, F., Turnbull, D. (Eds.), Solid State Physics, vol. 24,pp. 1–36.

Gubbins, D., Masters, G., 1979. Driving mechanisms for theEarth’s dynamo. Adv. Geophys. 21, 1–50.

Gubbins, D., Alfè, D., Masters, G., Price, G.D., Gillan, M.J., 2003.Can the Earth’s dynamo run on heat alone? Geophys. J. Int.,in press.

Heine, V., Weaire, D., 1970. Pseudopotential theory of cohesionand structure. In: Ehrenreich, H., Seitz F., Turnbull, D. (Eds.),Solid State Physics, vol. 24, pp. 249–463.

Hohenberg, P., Kohn, W., 1964. Inhomogeneous electron gas. Phys.Rev. B 136, 864–871.

Kohn, W., Sham, L.J., 1965. Self consistent equations includingexchange and correlation effects. Phys. Rev. A140, 1133–1138.

Jeanloz, R., 1979. Properties of iron at high pressures and thestate of the core. J. Geophys. Res. 84, 6059–6069.

Kresse, G., Furthermüller, J., 1996. Efficient iterative schemes forab-initio total-energy calculations using a plane-wave basis set.Phys. Rev. B54, 11169–11186.

Kresse, G., Joubert, D., 1999. From ultra soft pseudopotentials tothe projector augmented-wave method. Phys. Rev. B59, 1758–1775.

Kresse, G., Furthmüller, J., Hafner, J., 1995. Ab-initio force-constant approach to phonon dispersion relations of diamondand graphite. Europhys. Lett. 32, 729–734.

Laio, A., Bernard, S., Chiarotti, G.L., Scandolo, S., Tosatti, E.,2000. Physics of iron at Earth’s core conditions. Science 287,1027–1030.

Liu, L., Bassett, W.A., 1986. Elements, Oxides, Silicates: High-Pressure Phases with Implications for the Earth’s Interior. OUP,New York.

Mao, H.K., Shu, J., Shen, G., Hemley, R.J., Li, B., Sing, A.K.,1999. Elasticity and rheology of iron above 220 GPa and thenature of the Earth’s inner core. Nature 399, 280.

Mao, H.K., Xu, J., Struzhkin, V.V., Shu, J., Hemley, R.J., Sturhahn,W., Hu, M.Y., Alp, E.E., Vocadlo, L., Alfè, D., Price, G.D.,Gillan, M.J., Schwoerer-Bohning, M., Hausermann, D., Eng,P., Shen, G., Giefers, H., Lubbers, R., Wortmann, G., 2001.Phonon density of states of iron up to 153 gigapascals. Science292, 914–916.

Matsui, M., Anderson, O.L., 1997. The case for a body-centeredcubic phase for iron at inner core conditions. Phys. Earth Planet.Interiors 103, 55–62.

Matsui, M., Price, G.D., Patel, A., 1994. Comparison between thelattice dynamics and molecular dynamics methods: calculationresults for MgSiO3 perovskite. Geophys. Res. Lett. 21, 1659–1662.

McDonough, W.F., Sun, S.-S., 1995. The composition of the Earth.Chem. Geol. 120, 223–253.

Mukherjee, S., Cohen, R.E., 2002. Tight-binding based non-collinear spin model and magnetic correlations in iron. J.Computer-Aided Mater. 8, 107–115.

Nguyen, J.H., Holmes, N.C., 1999. Iron sound velocities in shockwave experiments. Shock Compres. Condens. Matter CP505,81–84.

Parker, S.C., Price, G.D., 1989. Computer modelling of phasetransitions in minerals. Adv. Solid State Chem. 1, 295–327.

Parrinello, M., Rahman, A., 1980. Crystal structure and pairpotentials: a molecular dynamics study. Phys. Rev. Lett. 45,1196–1199.

Poirier, J.P., 1994a. Light elements in the Earth’s outer core: acritical review. Phys. Earth Planet. Inter. 85, 319–337.

Poirier, J.P., 1994b. Physical-properties of the Earths core. C.R.Acad. Sci. II 318, 341–350.

Pulay, P., 1980. Convergence acceleration of iterative sequences.The case of SCF iteration. Chem. Phys. Lett. 73, 393.

Saxena, S.K., Dubrovinsky, L.S., Häggkvist, P., 1996. X-rayevidence for the new phase of�-iron at high temperature andhigh pressure. Geophys. Res. Lett. 23, 2441–2444.

Shen, G.Y., Heinze, D.L., 1998. High Pressure melting of deepmantle and core materials. In: Hemley, R.J. (Ed.), Ultrahigh-Pressure Mineralogy. Reviews in Mineralogy, vol. 37, pp. 369–398.


Shen, G.Y., Mao, H.K., Hemley, R.J., Duffy, T.S., Rivers, M.L.,1998. Melting and crystal structure of iron at high pressuresand temperatures. Geophys. Res. Lett. 25, 373.

Söderlind, P., Moriarty, J.A., Wills, J.M., 1996. First-principlestheory of iron up to earth-core pressures: structural,vibrational and elastic properties. Phys. Rev. B53, 14063–14072.

Steinle-Neumann, G., Stixrude, L., Cohen, R.E., 1999. First-principles elastic constants for the hcp transition metals Fe, Co,and Re at high pressure. Phys. Rev. B60, 791–799.

Steinle-Neumann, G., Stixrude, L., Cohen, R.E., Gülseren, O.,2001. Elasticity of iron at the temperature of the Earth’s innercore. Nature 413, 57–60.

Stixrude, L., Brown, J.M., 1998. The Earth’s core. In: Hemley, R.J.(Ed.), Ultrahigh-Pressure Mineralogy. Reviews in Mineralogy,vol. 37, pp. 261–283.

Stixrude, L., Cohen, R.E., 1995. Constraints on the crystallinestructure of the inner core—mechanical instability of bcc ironat high-pressure. Geophys. Res. Lett. 22, 125–128.

Stixrude, L., Wasserman, E., Cohen, R.E., 1997. Composition andtemperature of the Earth’s inner core. J. Geophys. Res. 102,24729–24739.

Stixrude, L., Cohen, R.E., Hemley, R.J., 1998. Theory of mineralsat high pressure. In: Hemley, R.J. (Ed.), Ultrahigh-PressureMineralogy. Reviews in Mineralogy, vol. 37, pp. 639–671.

Vanderbilt, D., 1990. Soft self-consistent pseudopotentials in ageneralized eigenvalue formalism. Phys. Rev. B41, 7892–7895.

Vocadlo, L., Alfè, D., 2002. The ab initio melting curve ofaluminum. Phys. Rev. B65 (214105), 1–12.

Vocadlo, L., deWijs, G.A., Kresse, G., Gillan, M.J., Price, G.D.,1997. First principles calculations on crystalline and liquid ironat Earth’s core conditions. Faraday Dis. 106, 205–217.

Vocadlo, L., Brodholt, J., Alfè, D., Price, G.D., Gillan, M.J., 1999.The structure of iron under the conditions of the Earth’s innercore. Geophys. Res. Lett. 26, 1231–1234.

Wang, Y., Perdew, J.P., 1991. Correlation hole of the spin-polarizedelectron-gas, with exact small-wave-vector and high-densityscaling. Phys. Rev. B44, 13298–13307.

Wasserman, E., Stixrude, L., Cohen, R.E., 1996. Thermalproperties of iron at high pressures and temperatures. Phys.Rev. B53, 8296–8309.

Yoo, C.S., Holmes, N.C., Ross, M., Webb, D.J., Pike, C., 1993.Shock temperatures and melting of iron at Earth core conditions.Phys. Rev. Lett. 70, 3931–3934.

Documents

The properties of iron under core conditions from ﬁrst principles … · 2012-03-26 · Physics of the Earth and Planetary Interiors 140 (2003) 101–125 The properties of iron